Frequent points for random walks in two dimensions

2006/07/25 by Richard F. Bass, Bass, Richard F., Jay Rosen +1
Computer Science · Mathematics · #60J65 #Bayesian Methods and Mixture Models #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.math/0607636

openalex publication_date 2006/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a symmetric random walk in Z² which does not necessarily have bounded jumps we study those points which are visited an unusually large number of times. We prove the analogue of the Erdős-Taylor conjecture and obtain the asymptotics for the number of visits to the most visited site. We also obtain the asymptotics for the number of points which are visited very frequently by time n. Among the tools we use are Harnack inequalities and Green's function estimates for random walks with unbounded jumps; some of these are of independent interest.

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